Interpretive Summary: Mathematical models are useful tools for investigating contaminant migration in soils and groundwater. Models, for example, can be used to estimate the amount of chemical that will reach drinking water wells if the chemical is applied to the soil surface. Because of the way soils are formed, most soils have layers (called horizons), with each layer having different physical and chemical characteristics. Accounting for the different layer properties is important for developing accurate models. In this work, we formulated a new mathematical model for chemical transport in layered materials such as soils. The model will allow for more realistic simulations of chemical transport in soils. The research will benefit scientists and engineers seeking to prevent or remediate groundwater and soils contamination.

Technical Abstract:
The advection-dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection-diffusion eigenvalue problem that had the same form and coefficients as the original problem. The generalized solution of the eigenvalue problem for any numbers of layers was developed using mathematical induction, establishing recurrence formulas and a transcendental equation for determining the eigenvalues. The orthogonality property of the eigenfunctions was found using an integrating factor that transformed the non-self-adjoint advection-diffusion eigenvalue problem into a purely diffusive, self-adjoint problem. The performance of the closed-form analytical solution was evaluated by solving the advection-dispersion transport equation for two- and five-layer media test cases which have been previously reported in the literature. Additionally, a solution featuring first-order decay was developed. The analytical solution reproduced results from the literature, and it was found that the rate of convergence for the current solution was superior to that of previously published solutions.